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A figurate number, also (but mostly in texts from the 1500 and 1600s) known as a figural number (Simpson and Weiner 1992, p. 587), is a number that can be represented by a ...

A Pascal's triangle written in a square grid and padded with zeros, as written by Jakob Bernoulli (Smith 1984). The figurate number triangle therefore has entries a_(ij)=(i; ...

The word "number" is a general term which refers to a member of a given (possibly ordered) set. The meaning of "number" is often clear from context (i.e., does it refer to a ...

A figurate number which is given by Ptop_n=1/4Te_n(n+3)=1/(24)n(n+1)(n+2)(n+3), where Te_n is the nth tetrahedral number. The first few pentatope numbers are 1, 5, 15, 35, ...

A figurate number of the form g_n=2n-1 giving the area of the square gnomon obtained by removing a square of side n-1 from a square of side n, g_n = n^2-(n-1)^2 (1) = 2n-1. ...

A number of the form Tt_n=((n+2; 2); 2)=1/8n(n+1)(n+2)(n+3) (Comtet 1974, Stanley 1999), where (n; k) is a binomial coefficient. The first few values are 3, 15, 45, 105, 210, ...

A figurate number of the form 4n^2-3n. The first few are 1, 10, 27, 52, 85, ... (OEIS A001107). The generating function giving the decagonal numbers is ...

A figurate number which is equal to the cubic number n^3. The first few are 1, 8, 27, 64, ... (OEIS A000578).

A nexus number is a figurate number built up of the nexus of cells less than n steps away from a given cell. The nth d-dimensional nexus number is given by N_d(n) = ...

A figurate number in which layers of polygons are drawn centered about a point instead of with the point at a polygon vertex.